 11.3 (Hausdorff measures) 
Here, we briefly comment on why the lower densities set up so far in
this section do not help to describe explicitly the cccc and lld
version of the saturation of the MSN
\((\mathbb {R}^m,{B}(\mathbb {R}^m),{N}_{{H}^k})\) . The main reason is that we would
need to enlarge the collection \({E}\)  for it to be generating, since
there are (much) less \({H}^k\) -negligible sets than there are purely
\(k\) -unrectifiable sets. In doing so we loose (REF ),
which was critical for implementing the techniques of the previous
section. In fact, if \(M \subseteq \mathbb {R}^m\)  is Borel, \(\phi _M = {H}^k \hspace{2.5pt}{\vrule height7pt width.5pt depth0pt}\hspace{-0.2pt}\vbox {\hrule height.5pt width7pt depth0pt}\, M\) 
is locally finite, and \(\Theta ^k(\phi _M,x)=1\)  for \({H}^k\) -almost
every \(x \in M\) , then \(M\)  is countably \(({H}^k,k)\) -rectifiable, see
e.g. [1]}. Since we ought to include non
\({H}^k\) -negligible, purely \(k\) -unrectifiable sets in an
\({N}_{{H}^k}\) -generating family, our only choice is, if possible,
to change the definition of the lower densities \(\Theta _M\) . So far, we
do not know how to construct, in this case, a compatible family of
lower densities.

